Let us now study the combined phenomenological consequences of \(Z-Z^{\prime }\)  mixing in \(b\rightarrow s\ell ^+\ell ^-\)  and the global EW fit. For this purpose we will focus on an illustrative simplified scenario for an \(SU(2)_L\)  singlet \(Z^\prime \) . As discussed in the previous section, \(b\rightarrow s\ell ^+\ell ^-\)  data motivates vectorial couplings to leptons, i.e. \(g_{ii}^{\ell L} = g_{ii}^{\ell R} = g_{ii}^{\ell V}\)  which also allow for simple configurations without gauge anomalies such as \(L_\mu -L_\tau \)  [1]}, [2]} or \(B_3-L_2\)  [3]}. We consider in this illustrative scenario only left-handed \(Z^{\prime }-b-s\)  couplings but assume absence of sizable couplings to light quarks in order to avoid direct LHC searches. Note that such a scenario could be generated in models with vector-like quarks [1]}, [5]},. While these assumptions are justified by the \(b\rightarrow s\ell ^+\ell ^-\)  fit, we will assume in addition \(g_{11}^{\ell V}=0\)  and \(g_{22}^{\ell V}=-g_{33}^{\ell V}=g^\prime \) , i.e. a \(L_\mu =-L_\tau \)  symmetry [6]}, [7]}, [8]}. This does not only avoid the LEP II bounds on 4-lepton contact interactions but also the effect of \(Z-Z^\prime \)  mixing in the total \(Z\)  width will cancel to leading order such that in this scenario the largest lepton couplings are possible (see Fig. REF ) and \(\tau \rightarrow \mu \nu \nu \)  receives the desired constructive contribution via \(W-Z^\prime \)  box diagrams.Note, that our analysis would to a good approximation also apply to other scenarios, such as \(B_3-L_2\) .
